\newproblem{lay:4_2_10}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.2.10}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Ana Pe\~na Gil, Jan. 19th 2014} \\}{}

  % Problem statement
	For the set below, either find an appropriate theorem to show that $W$ is a vector space or
	find a specific example to show the contrary.
	\begin{center}
		$W=\left\{\begin{pmatrix}a\\b\\c\\d\end{pmatrix}\left|3a+b=c, a+b+2c=2d\right.\right\}$
	\end{center}
}{
  % Solution
	We can rewrite the two conditions for the vectors in $W$ as
	\begin{center}
	   $\begin{pmatrix}3 & 1 & -1 & 0 \\ 1 & 1 & 2 & -2\end{pmatrix}\begin{pmatrix}a\\b\\c\\d\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$
	\end{center}
	So, $W$ is nothing more than the null space of the matrix $A=\begin{pmatrix}3 & 1 & -1 & 0 \\ 1 & 1 & 2 & -2\end{pmatrix}$ and consequently it is a vector
	subspace of $\mathbb{R}^4$. Since any vector subspace is a vector space, then $W$ is a vector space.\\
}
\useproblem{lay:4_2_10}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
